Question

Consider the partial differential equation:

∂u/∂x + 2∂u/∂t + αu = 0, -[infinity]0 is a constant, with boundary condition u(0, t) = f(t).

Answer 1

The given equation is a partial differential equation. To solve it, we can use the method of separation of variables. By assuming the solution is u(x, t) = X(x)T(t) and applying appropriate substitutions and boundary conditions, we can find the general solution as u(x, t) = (f(t) cos(√(λ² + α)x) + B sin(√(λ² + α)x))e^(-2λt).

Step-by-step explanation:

The given equation ∂u/∂x + 2∂u/∂t + αu = 0 is a partial differential equation.

To solve this equation, we can use the method of separation of variables.

1. Assume that the solution can be expressed as a product of two functions u(x, t) = X(x)T(t).
2. Substitute this assumption into the equation and divide by X(x)T(t).
3. Since the left side of the equation depends only on x and the right side depends only on t, the only way for this equation to hold for all x and t is if both sides are equal to a constant. Let's call this constant -λ².
4. This gives us two ordinary differential equations: X''(x) + αX(x) = -λ²X(x) and T'(t) + 2λT(t) = 0.
5. Solve each of these equations separately.
6. The general solution to the first equation is X(x) = A cos(√(λ² + α)x) + B sin(√(λ² + α)x).
7. The general solution to the second equation is T(t) = Ce^(-2λt).
8. Combining X(x) and T(t), we get u(x, t) = (A cos(√(λ² + α)x) + B sin(√(λ² + α)x))e^(-2λt).
9. Apply the boundary condition u(0, t) = f(t) to determine the values of A and B.
10. The final solution is u(x, t) = (f(t) cos(√(λ² + α)x) + B sin(√(λ² + α)x))e^(-2λt), where B can be determined from the boundary condition.